diagnosi automatica dei guasti - silvio simani

Tecniche di Diagnosi Tecniche di Diagnosi Automatica dei GuastiAutomatica dei Guasti

Parte 1Parte 1

Silvio SimaniMarch 2011

URL: www.silviosimani.itE-mail: [email protected]

http://www.silviosimani.it/

mailto:[email protected]

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Tecniche di Diagnosi Automatica deiTecniche di Diagnosi Automatica dei GuastiGuasti. . Silvio SimaniSilvio Simani2

Lecture Main TopicsGeneral introductionGeneral introduction

StateState--ofof--thethe--art reviewart reviewFault diagnosis nomenclatureFault diagnosis nomenclature

Main methods for fault diagnosisMain methods for fault diagnosisParameter estimation methodsParameter estimation methodsObserver and filter approachesObserver and filter approachesParity relationsParity relationsNeural networks and fuzzy systemsNeural networks and fuzzy systems

Application examplesApplication examplesConcluding remarksConcluding remarks

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Programme DetailsIntroduction: Course Introduction Issues in Model-Based Fault Diagnosis Fault Detection and Isolation (FDI) Methods based on Analytical Redundancy Model-based Fault Detection Methods The Robustness Problem in Fault Detection Fault Identification Methods Modelling of Faulty Systems

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Programme Details (Cont’d)

Residual Generation Techniques The Residual Generation Problem

Fault Diagnosis Technique Integration Fuzzy Logic for Residual Generation Neural Networks in Fault Diagnosis

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Programme Details (Cont’d)Output Observers for Robust Residual Generation Unknown Input Observer (UIO) FDI Schemes Based on UIO and Output Observers Kalman Filtering and FDI from Noisy Measurements Residual Robustness to Disturbances

Application Examples

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Annual Meetings on FDIAnnual Meetings on FDIIFAC SAFEPROCESS SymposiumSymposium on Fault Detection Supervision and Safety for Technical Processes

1st held in Baden–Baden, Germany in 19912nd in Espo, Finland in 19943rd at Hull, UK in 19974th held in Budapest, Hungary in 20005th at Washington DC in July 20036th in Beijing, P.R. China, August 20067th in Barcelona, Spain, July 20098th scheduled in Mexico City, Mexico, August 2012

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Nomenclature

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Nomenclature (cont’d)

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Nomenclature (Cont’d)

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NOTE: Incipient fault (slowly developing fault)= hard to detect NOTE: Incipient fault (slowly developing fault)= hard to detect !!!!!!

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Application Examples

Simulated case studiesSimulated case studies

Identification/FDI applicationsIdentification/FDI applications

Real processesReal processes

Research worksResearch works

Undergraduate theses topics Undergraduate theses topics

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Simulated Application ExamplesSimulated Gas TurbineSimulated Gas Turbine

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Simulated Application Examples (Cont’d)Simulated Gas TurbineSimulated Gas Turbine

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Simulated Application Examples (Cont’d)Simulated Gas Turbine (SIMULINKSimulated Gas Turbine (SIMULINK®®))

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Simulated Application Examples (Cont’d)Simulated Power Plant: Simulated Power Plant: Pont sur Pont sur SambreSambre

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Simulated Application Examples (Cont’d)Small Aircraft ModelSmall Aircraft Model

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Simulated Application Examples (Cont’d)

Small Aircraft ModelSmall Aircraft Model

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Aerospace SatelliteAerospace Satellite

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Simulated Application Examples (Cont’d)Manifacturing ProcessManifacturing Process

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Manifacturing ProcessManifacturing Process JAM!

(possibly unskilled)human operator

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Introduction

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Introduction (Cont’d)

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Residual GenerationThis block generates residual signals using available inputs and outputs from the monitored systemThis residual (or fault symptom) should indicate that a fault has occurredNormally zero or close to zero under no fault condition, whilst distinguishably different from zero when a fault occurs

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Residual EvaluationThis block examines residuals for the likelihood of faults and a decision rule is then applied to determine if any faults have occurredIt may perform a simple threshold test (geometrical methods) on the instantaneous values or moving averages of the residualsIt may consist of statistical methods, e.g., generalised likelihood ratio testing or sequential probability ratio testing

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Introduction (Cont’d)ModelModel--Based FDI Methods:Based FDI Methods:

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Introduction (Cont’d)Signal ModelSignal Model--Based Methods:Based Methods:

Change Detection: Residual AnalysisChange Detection: Residual Analysis

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Introduction (Cont’d)Model Uncertainty and FDI

Model-reality mismatchSensitivity problem: incipient faults!

Robustness in FDIDisturbance, modelling errors, uncertaintyUIO and Kalman filter: robust residual generation

System Identification for FDIEstimation of a reliable modelModelling accuracyDisturbance estimation (recall: ARX, ARMAX, BJ)

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Introduction (Cont’d)Fault Identification MethodsFault Identification Methods

Fault nature (type, shape) & size (amplitude)Fault nature (type, shape) & size (amplitude)

Approximate Reasoning Methods:Approximate Reasoning Methods:

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Introduction (Cont’d)FDI applications status & reviewFDI applications status & review

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Model-based FDI Techniques

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ModelModel--based FDI Techniques (Cont’d)based FDI Techniques (Cont’d)

1.1.

2.2.

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Modelling of Faulty Systems

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Fault Location:Fault Location:ActuatorsActuatorsProcess or system Process or system componentscomponentsInput sensorsInput sensorsOutput sensorsOutput sensorsControllersControllers

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Fault and System Fault and System ModellingModelling

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ModelModel--based FDI Techniques (Cont’d)based FDI Techniques (Cont’d)Fault and System Fault and System ModellingModelling

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⎪⎩

⎪⎨

⎧

+=

+=++=+

)()()()()()(

)()()()1(

tttttt

tttt

uR

y

Rc

fuufxCy

uBfxAx

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Modelling Modelling of Faulty of Faulty SystemsSystems

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Modelling Modelling of Faulty of Faulty SystemsSystems

Transfer function Transfer function description:description:

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Residual Generator StructureResidual Generator Structure

Under faultUnder fault--freefreeassumptions, theassumptions, theresidual signal residual signal r(t)r(t)is is ““almost zeroalmost zero””

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Residual General Structure (Cont’d)Residual General Structure (Cont’d)

Residual generationResidual generationvia system simulatorvia system simulator

z(t) is the simulatedplant output

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Residual General Structure (Cont’d)Residual General Structure (Cont’d)

Residual generator:Residual generator:

Constraint conditions: Constraint conditions: designdesign

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General Residual EvaluationGeneral Residual Evaluation

Faulty residual

Faultfreeresidual

Detection thresholdsε(t)

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General Residual Evaluation General Residual Evaluation (example)(example)

Fault free residualFault free residual FaultFault--free & free & faultyfaulty

residualsresiduals

Detection thresholdsDetection thresholds

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Residual Generation TechniquesResidual Generation Techniques

Fault detection via parameter

estimation

Observer-based approaches

Parity (vector) relations

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Fault Detection via Fault Detection via Parameter EstimationParameter Estimation

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Parameter Estimation Parameter estimation for fault detectionThe process parameters are not known at all, or they are not known exactly enough. They can be determined with parameter estimation methodsThe basic structure of the model has to be known Based on the assumption that the faults are reflected in the physical system parametersThe parameters of the actual process are estimated on-line using well-known parameter estimations methods

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Parameter Estimation (Cont’d)The results are thus compared with the parameters of the reference model obtained initially under fault-free assumptionsAny discrepancy can indicate that a fault may have occurred

An approach for modelling the input-output behaviour of the monitored system will be recalled and exploited for fault detection

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Equation Error Approach

SISO model

Parameter vector

Regression vector

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Equation Error Method

Equation error

Model of the process (Z-transform)

Estimated polynomials

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Equation Error Method (Cont’d)

Estimation of the process model: LS

LS minimisation

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Equation Error Method (Cont’d)

Estimation of the process model: RRLS

Estimate recursive adaptation[ ])(ˆ)()()()(ˆ)1(ˆ tttyttt T ΘΨΘΘ −+=+ γ

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Parameter Estimation via EERecursive estimation of the transfer function polynomials

Equation errorParameter estimation via recursive algorithm

RLSRLS

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Links Between InputLinks Between Input-- Output and StateOutput and State--Space Space

DiscreteDiscrete--Time LTI ModelsTime LTI Models

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Input-Output Model (EE)The input to output discrete-time model behaviour can be mathematically described by a set of ARX Multi-Input Single-Output (MISO) models

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Input-Output Model (EE)m is the number of the output variables. The order n and the parameters αi,j and βi,j,k with i = 1,…,m, of the model are determined by the identificationapproach. The term εi(t) takes into account the modelling error (EE)

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State-Space Equivalent ModelThe input-output EE model has a state space “realisation” as follows:

The matrices (Ai, Bi, Ci, Bωι, Dωι

) of a state space representation in canonical form of the n-th order system are deefined as follows:

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State-Space Matrices

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State-Space Matrices (Cont’d)

Note that theNote that thematrix matrix SSii is always is always nonnon--singular singular

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State-Space Matrices (Cont’d)TF2SSTF2SS Transfer function to state-space conversion

[A,B,C,D] = TF2SS(NUM,DEN)[A,B,C,D] = TF2SS(NUM,DEN) calculates the state- space representation (A,B,C,D) of the system (NUM,DEN) from a single input. Vector DEN must contain the coefficients of the denominator in descending powers of s (z). Matrix NUM must contain the numerator coefficients with as many rows as there are outputs y. The A,B,C,D matrices are returned in controller canonical form. This calculation also works for discrete systems

For MIMOMIMO models, use ssss and tftf Matlab functions

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ObserverObserver--based based ApproachesApproaches

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Residual General StructureResidual General StructureObserverObserver--based approachbased approach

Plant modelPlant model

Observer modelObserver model

Output estimation approach!Output estimation approach!

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Residual Generator StructureResidual Generator Structure

Plant modelPlant model

Observer modelObserver model

State estimationState estimation modelmodel

State estimationState estimation propertyproperty (fault(fault--free case!!!)free case!!!)

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Residual Generator PropertyResidual Generator Property+ disturbance signals and fault + disturbance signals and fault

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Residual Generator Property (Cont’d)Residual Generator Property (Cont’d)

+ fault signals + fault signals

System model System model

Observer model Observer model

Output estimation errorOutput estimation errorwith faults with faults but noisebut noise--free free

Both Both e(t)e(t) and and eexx (t) (t) are suitable residuals!are suitable residuals!

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Output Observer & UIO: Output Observer & UIO: introintroState transformation

Observer

State estimation error

Residual

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General Residual EvaluationGeneral Residual Evaluation

Faulty residual

Faultfreeresidual


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Change Detection & Residual EvaluationChange Detection & Residual Evaluation

Faulty residual

Faultfreeresidual


3with),,1()(

≥=×±=

δσδε mirt ii

( ) )()( trtrJ ≡

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Residual GenerationResidual Generation

Output ObserversOutput ObserversUnknown Input ObserversUnknown Input ObserversFault DetectionFault Detection

Fault IsolationFault Isolation, , i.e. where is the i.e. where is the faultfault? ?

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Output Observer

Process model with faultsProcess model with faults

InputInput--output sensor faultsoutput sensor faults

Observer for the iObserver for the i--th output yth output yii (t)(t)

( ) modelprocessspacestatetheis,, −iii CBA

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Output Observer for Fault Detection

Given the observer modelGiven the observer model

Under faultUnder fault--free assumptionsfree assumptions

Fault detection logicFault detection logic

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Output Observer for Fault Isolation

Process model

Bank of output observersBank of output observers

)()()( * tftyty ii +=

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Output Observer for Fault Isolation (Cont’d)(Cont’d)Bank of Bank of output observersoutput observers

)()()( * tftyty ii +=

FaultFault--free case:free case:

Faulty caseFaulty case

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Output Observer for Fault Isolation (Cont’d)(Cont’d)Bank of Bank of output observersoutput observers

)()()( * tftyty ii +=

FaultFault--free case:free case:

Faulty caseFaulty case0)(lim ≠

∞→trit

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Unknow Input Observer (UIO)

System with disturbanceSystem with disturbance““Unknown InputUnknown Input””

DefinitionDefinition:: An observer is defined as an An observer is defined as an Unknown Input ObserverUnknown Input Observer forfor the system the system with disturbance (above),with disturbance (above), if its state estimation if its state estimation

error vector error vector eexx (t)(t) approaches zero asymptotically, regardless of the approaches zero asymptotically, regardless of the presence of the unknown inputpresence of the unknown input term in the system.term in the system.

Unknown Unknown InputInput

Disturbance Distribution Matrix Disturbance Distribution Matrix (Known)(Known)

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UIO Model

Given:Given:

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UIO Structure

Plant ModelPlant Model

UIO ModelUIO Model

Observer Design???Observer Design???

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UIO DesignPlant ModelPlant Model

UIO ModelUIO Model

)1(ˆ)1()1( +−+=+ txtxtex State estimation errorState estimation error

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UIO Design (Cont’d)(Cont’d)

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This means that, if all the eigenvalues of F are stable, ex (t) will approach zero asymptotically, i.e., . Hence, according to the definition of the UOI described by the model above, it is an UIO for the monitored system. The design of this UIO consists of solving the equation system and making all eigenvalues of the system matrix F be stable.

)()(ˆ txtx →

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******

******

**

**********

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UIO for Fault (Detection) + Isolation

General systemGeneral system

UIO modelUIO model

Process with input faultsOutput sensorsOutput sensorsare faultare fault--freefree

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UIO for FDI

UIO for Residual GenerationUIO for Residual Generation

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UIO for FDI (Cont’d)

UIO for Residual GenerationUIO for Residual Generation

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UIO for inputUIO for inputsensor sensor ffuu (t) (t) fault isolationfault isolation ⎪⎭

⎪⎬⎫

=

+=+

)()(

)()()1(

1 tt

tttix

iiu

iix

iix

eLr

fBTeFe

⎪⎭

⎪⎬⎫

=

+=+

)()(

)()()1(

1 tt

tttix

iiu

iix

iix

eLr

fBTeFe

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UIO for FDI (Cont’d) * **

*

**

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*

*

**

**

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UIO for FDI (Cont’d) Σ

ΣΣ

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Fault Isolation with UIOEach observer is insensitive toEach observer is insensitive toone input sensor: one input sensor:

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Multiple FDI

Input sensor FDIInput sensor FDI Input sensor FDIInput sensor FDI

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Multiple FDI (Cont’d)

Fault SignatureFault Signature

UIOUIO

DynamicDynamicObserversObservers

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Multiple FDI (Cont’d)

i)

ii)

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Residual Disturbance Robustness

Residuals Residuals decoupled from decoupled from disturbancedisturbanceRobust residual Robust residual generatorgeneratorDisturbance Disturbance effect effect minimisationminimisationMeasurement Measurement errorserrors

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FDI with Noisy Measurements

Model with fault and noiseModel with fault and noise

Model with noise only: Model with noise only: Kalman filterKalman filter!!

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Kalman Filter Design

(State and Output Prediction)(State and Output Prediction)

*

*

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Kalman Filter Design (Cont’d)

(Vector Updates)(Vector Updates)

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Defined:It minimises: i.e. the mean square error & the error covariance matrix

Filter gainSolution of the difference equation:

Kalman Filter Properties (Cont’d)

)(ˆ)()( txtxte −=[ ] )()()( tPteteT ≡Ε

)(tK

[ ]QAtPA

AtPCCtPCRCtPAtP TTT

++++−=+

−

)()()()()1(

1

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Kalman Filtering for FDI

*

**

1)1)

2)2)

3)3)

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Kalman Filtering for FDI (Cont’d)

)|1()1()1()1()1( ttxCtytytyte iFii

iFii +−+=+−+=+InnovationInnovation

(i)(i)

(ii)(ii)

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KF Residual Evaluation...KF Residual Evaluation...



Which thresholds???Which thresholds???

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)|1()1()1()1()1()( ttxCtytytytetr iFii

iFii +−+=+−+=+≡

Innovation orInnovation orResidual Residual r(t)r(t)

(i)(i) Statistical TestsStatistical Tests

ZeroZero--meanmean

&&

variancevariance

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)|1()1()1()1()1()( ttxCtytytytetr iFii

iFii +−+=+−+=+≡

Innovation orInnovation orResidual Residual r(t)r(t)

(ii)(ii) Statistical TestsStatistical Tests

Whiteness testWhiteness test

χχ22 −− typetype

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Whiteness testWhiteness test

χχ22 −− typetype

previously seen

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KF Residuals: Mean-value (example)

FaultFault--freefree&&

faultyfaultyresidualsresiduals

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KF Residuals: Standard deviation (example)



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KF Residuals: Whiteness test (example)



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Fault Detection with Fault Detection with Parity EquationsParity Equations

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Parity Relations for Fault DetectionThe basic idea of the parity relations approach is to provide a proper check of the parity (consistency) of the measurements acquired from the monitored systemIn the early development of fault diagnosis, the parity vector (relation) approach was applied to static or parallel redundancy schemes, which may be obtained directly from measurements(hardware redundancy) or from analytical relations (analytical redundancy)

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Parity Relations for Fault DetectionIn the case of hardware redundancy, two methods can be exploited to obtain redundant relationsThe first requires the use of several sensors having identical or similar functions to measure the same variableThe second approach consists of dissimilar sensors to measure different variables but with their outputs being relative to each other

Analytical forms of redundancyAnalytical forms of redundancy

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Analytical Redundancy

Model (M) and process (P)

Error vector

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Analytical Redundancy (Cont’d)

Model (M) and process (P)

Error vector

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Analytical Redundancy (EEEE)

Models, if:

Residual, with input and output faults

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Analytical Redundancy (OEOE)

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Parity Relation (EEEE)

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Parity Relations via EEEEAccording to EE, another possibility forgenerating a polynomial error:

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Parity RelationsThe previous equations that generate residuals and are called parity equations under the assumptions of fault occurrence and of exact agreement between process and modelHowever, within the parity equations, the model parameters are assumed to be known and constant, whereas the parameter estimationsparameter estimationscan vary the parameters of the polynomials in order to minimise the residuals

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Change Detection and Change Detection and Symptom EvaluationSymptom Evaluation

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Residual Evaluation

When the residual generation stage has been performed, the second step requires the examination of symptoms in order to determine if any faults have occurredA decision process may consist of a simplethreshold test on the instantaneous values of moving averages of residuals

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Residual Evaluation (Cont’d)On the other hand, because of the presence of noise, disturbances and other unknown signals acting upon the monitored system, the decision making process can exploits statistical methodsIn this case, the measured or estimated quantities, such as signals, parameters, state variables or residuals are usually represented by stochastic variables

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Residual Evaluation (Cont’d)Mean and standard values

Residuals or symptoms

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Residual Evaluation (Cont’d)Fixed threshold selection

By a proper choice of ε, a compromise has to be made between the detection of small faults and false alarmsMore complex residual evaluation schemes

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Residual Generation Problem

Robustness issues!Robustness issues!Two design principles:

Uncertainty is taken into account at the residual design stage: active robustness in fault diagnosisactive robustness in fault diagnosis

Passive robustness Passive robustness makes use of a residual makes use of a residual evaluator with proper threshold selection evaluator with proper threshold selection methods (fixed or adaptive)methods (fixed or adaptive)

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Fault Diagnosis Fault Diagnosis Technique Technique

IntegrationIntegration

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FDI Technique IntegrationSeveral FDI techniques have been developed and their application shows different properties with respect of the diagnosis of different faults in a processTo achieve a reliable FDI technique, a good solution consists of a proper integration of several methods which take advantages of the different proceduresExploit a knowledge-based treatment of all available analytical and heuristic information

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Fuzzy Logic for Residual Generation

Classical fault diagnosis model-based methods can exploit state-space of input-output dynamic models of the process under investigationFaults are supposed to appear as changes on the system state or output caused by malfunctions of the components as well as of the sensorsThe main problem with these techniques is that the precision of the process model affects the accuracy of the detection and isolation system as well as the diagnostic sensitivity

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Fuzzy Logic for Residual Generation (Cont’d)

The majority of real industrial processes are nonlinear and cannot be modelled by using a single model for all operating conditionsSince a mathematical model is a description of system behaviour, accurate modelling for a complex nonlinear system is very difficult to achieve in practiceSometimes for some nonlinear systems, it can be impossible to describe them by analytical equations

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Fuzzy Logic for Residual Generation (Cont’d)

Sometimes the system structure or parameters are not precisely known and if diagnosis has to be based primarily on heuristic information, no qualitative model can be set upBecause of these assumptions, fuzzy system theory seems to be a natural tool to handle complicated and uncertain conditionsInstead of exploiting complicated nonlinear models, it is also possible to describe the plant by a collection of local affine fuzzy models, whose parameters are obtained by identification procedures

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Residual Generation via Fuzzy Models

Resisual signals:Resisual signals:

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Neural Networks in Fault Diagnosis

Quantitative model-based fault diagnosis generates symptoms on the basis of the analytical knowledge of the process under investigationIn most cases this does not provide enough information to perform an efficient FDI, i.e., to indicate the location and the mode of the faultA typical integrated fault diagnosis system uses both analytical and heuristic knowledge of the monitored system

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Neural Networks in Fault Diagnosis (Cont’d)

The knowledge can be processed in terms of residual generation (analytical knowledge) and feature extraction (heuristic knowledge)The processed knowledge is then providedto an inference mechanism which can comprise residual evaluation, symptomobservation and pattern recognition

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In recent years, neural networks (NN) have been used successfully in pattern recognition as well as system identification, and they have been proposed as a possible technique for fault diagnosis, tooNN can handle nonlinear behaviour and partially known process because they learn the diagnostic requirements by means of the information of the training data

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NN are noise tolerant and their ability to generalise the knowledge as well as to adapt during use are extremely interesting properties FDI is performed by a NN using input and output measurements

NN is trained to identify the fault from measurement patternsClassification of individual measurement pattern is not always unique in dynamic situations, therefore the straightforward use of NN in

Fault diagnosis of dynamic plant is not practical and other approaches should be investigated

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A NN could be exploited in order to find a dynamic model of the monitored system or connections from faults to residualsIn the latter case, the NN is used as pattern classiifier or nonlinear function approximatorNN are capable of approximating a large class of functions for fault diagnosis of an industrial plantThe identification of models for the system under diagnosis as well as the application of NN as function approximator will be shown

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Quantitative and qualitative approaches have a lot of complementary characteristics which can be suitably combined together to exploit their advantages and to increase the robustness of quantitative techniquesPartial knowledge deriving from qualitative reasoning is reduced by quantitative methodsFurther research on model-based fault diagnosis consists of finding the way to properly combine these two approaches together to provide highly reliable diagnostic information

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FDI with Neural Networks

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FDI with Neural Networks (Cont’d)

As described in the figure, the fault diagnosis methodology consist of 2 stagesIn 1st stage, the fault has to be detected on the basis of residuals generated from a bank of output estimators, while, in the 2nd step, fault identification is obtained from pattern recognition techniques implemented via NNFault identification represents the problem of the estimation of the size of faults occurring in a dynamic system

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A NN is exploited to find the connection from a particular fault regarding system inputs and output measurements to a particular residualThe output predictor generates a residual which does not depend on the dynamic characteristics of the plant, but only on faultsNNs classify static patterns of residuals, which are uniquely related to particular fault conditions independently from the plant dynamics

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NNs have been used both as predictor of dynamic models for fault diagnosis, and pattern classifiers for fault identificationThe most frequently applied neural models are the feed-forward perceptron used in multi-layer networks with static structureThe introduction of explicit dynamics requires the feedback of some outputs through time delay units

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Alternatively to static structure, NN with neurons having intrinsic dynamic properties can be usedOn the other hand, NN can be effectivelyexploited for residual signal processing, which is actually a static patter recognition problem

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Fault signals create changes in several residuals obtained by using outputpredictors of the process under examinationA neural network is exploited in order tofind the connection from a particular fault regarding input and output measurements to a particular residual

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The predictors generate residuals independent of the dynamic characteristics of the plant anddependent only on sensors faultsTherefore, the neural network evaluates static patterns of residuals, which are uniquely related to particular fault conditions independently from the plant dynamics

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Conclusion – Lecture 1

ModelModel--Based FDIBased FDI

Analytical RedundancyAnalytical Redundancy

StateState--Space ModelsSpace Models

Residual GenerationResidual Generation

Unknown Input Observers UIOUnknown Input Observers UIO

Dynamic Observers / Kalman FiltersDynamic Observers / Kalman Filters

Residual Evaluation/Change DetectionResidual Evaluation/Change Detection

diagnosi automatica dei guasti - silvio simani

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